Average Error: 9.2 → 0.1
Time: 4.2s
Precision: 64
\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]
↓
\[\frac{x}{y} + \mathsf{fma}\left(2, \frac{\frac{1}{t}}{z}, 2 \cdot \frac{1}{t} - 2\right)\]
\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}
↓
\frac{x}{y} + \mathsf{fma}\left(2, \frac{\frac{1}{t}}{z}, 2 \cdot \frac{1}{t} - 2\right)
double f(double x, double y, double z, double t) {
        double r769946 = x;
        double r769947 = y;
        double r769948 = r769946 / r769947;
        double r769949 = 2.0;
        double r769950 = z;
        double r769951 = r769950 * r769949;
        double r769952 = 1.0;
        double r769953 = t;
        double r769954 = r769952 - r769953;
        double r769955 = r769951 * r769954;
        double r769956 = r769949 + r769955;
        double r769957 = r769953 * r769950;
        double r769958 = r769956 / r769957;
        double r769959 = r769948 + r769958;
        return r769959;
}
↓
double f(double x, double y, double z, double t) {
        double r769960 = x;
        double r769961 = y;
        double r769962 = r769960 / r769961;
        double r769963 = 2.0;
        double r769964 = 1.0;
        double r769965 = t;
        double r769966 = r769964 / r769965;
        double r769967 = z;
        double r769968 = r769966 / r769967;
        double r769969 = r769963 * r769966;
        double r769970 = r769969 - r769963;
        double r769971 = fma(r769963, r769968, r769970);
        double r769972 = r769962 + r769971;
        return r769972;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original9.2
Target0.1
Herbie0.1
\[\frac{\frac{2}{z} + 2}{t} - \left(2 - \frac{x}{y}\right)\]

Derivation

  1. Initial program 9.2

    \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\]

  2. Taylor expanded around 0 0.1

    \[\leadsto \frac{x}{y} + \color{blue}{\left(\left(2 \cdot \frac{1}{t \cdot z} + 2 \cdot \frac{1}{t}\right) - 2\right)}\]

  3. Simplified0.1

    \[\leadsto \frac{x}{y} + \color{blue}{\mathsf{fma}\left(2, \frac{1}{t \cdot z}, 2 \cdot \frac{1}{t} - 2\right)}\]
  4. Using strategy rm

  5. Applied associate-/r*0.1

    \[\leadsto \frac{x}{y} + \mathsf{fma}\left(2, \color{blue}{\frac{\frac{1}{t}}{z}}, 2 \cdot \frac{1}{t} - 2\right)\]

  6. Final simplification0.1

    \[\leadsto \frac{x}{y} + \mathsf{fma}\left(2, \frac{\frac{1}{t}}{z}, 2 \cdot \frac{1}{t} - 2\right)\]

Reproduce

herbie shell --seed 2020027 +o rules:numerics
(FPCore (x y z t)
  :name "Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2"
  :precision binary64

  :herbie-target
  (- (/ (+ (/ 2 z) 2) t) (- 2 (/ x y)))

  (+ (/ x y) (/ (+ 2 (* (* z 2) (- 1 t))) (* t z))))